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ASSIGNMENT 5 There are two parts to this assignment. The first part consists of 15 questions on WeBWorK — the link is available on the course webpage. The second part consists of the questions on this page. You are expected to provide full solutions with complete justifications. You will be graded on the mathematical, logical and grammatical coherence and elegance of your solutions. Your solutions must be typed, with your name and student number at the top of the first page. If your solutions are on multiple pages, the pages must be stapled together. Your written assignment must be handed in before the start of your recitation on Friday, March 6. The online assignment will close at 9:00 a.m. on Friday, March 6. 1. Determine whether the sequence {cos(n)} converges or diverges. 2. Given a sequence {an }, a subsequence {ank } is formed by selecting an infinite number of terms from that sequence (and preserving the order of terms). For example, {2, 3, 5, 7, 11, . . .} is a subsequence of {1, 2, 3, . . .}. Prove that every bounded sequence has a convergent subsequence. (Hint: construct the subsequence with the same method of bisection used last term on Assignment 7 to prove the Boundedness Theorem.) 3. Recall that the Fibonacci sequence {an } is defined by an = Let {rn } = n an+1 an o 1 an−1 + an−2 if n = 1, 2 . if n ≥ 3 be the sequence of ratios of subsequent terms in the Fibonacci sequence. In this question you will prove that {rn } converges to the golden ratio ϕ = √ 1+ 5 2 . (a) Explain why rn = 1 + 1 rn−1 for n ≥ 2 and rn = 2 − 1 rn−2 + 1 for n ≥ 3. (b) Explain why if rn−2 ≥ ϕ, then rn ≥ ϕ; and if rn−2 ≤ ϕ, then rn ≤ ϕ. (c) Prove that the sequence {r2n−1 } is increasing; and the sequence {r2n } is decreasing. (d) Prove that {rn } converges to ϕ.